Question

rationalising the denominator√7-√5/√7+√5​

Answer 1

Given:

• $\mathtt{ \frac{ \sqrt{7} - \sqrt{5} }{ \sqrt{7} + \sqrt{5} } }$

To Find:

• $\sf{Rationalisation \: of \: denominator.}$

Formulas Used:

• $\mathtt{( {a - b)}^{2} = {a}^{2} + {b}^{2} - 2ab}$

• $\mathtt{(a + b) (a - b) = {a}^{2} - {b}^{2} }$

Solution:

Let's rationalize the denominator,

$:\implies\mathfrak{ \frac{ \sqrt{7 - \sqrt{5} } }{ \sqrt{7} + \sqrt{5} } }$

$:\implies\mathfrak{ \frac{ \sqrt{7 - \sqrt{5} } }{ \sqrt{7} + \sqrt{5} } } \\ \\ \\ :\implies\mathfrak{ \frac{ {( \sqrt{7} - \sqrt{5}) }^{2} }{( { \sqrt{7)} }^{2} + { \sqrt{(5)} }^{2} } } \\ \\ \\ :\implies\mathfrak{ \frac{7 + 5 - 2 \sqrt{35} }{7 - 5} } \\ \\ \\ :\implies\mathfrak{ \frac{12 - 2 \sqrt{35} }{2} } \\ \\ \\ :\implies\mathfrak{ \frac{2(6 - \sqrt{35)} }{2} } \\ \\ \\ \boxed{:\implies\mathfrak\red{6 - \sqrt{35} }}$

Hence,

• The rationalization of denominator is 6 - √35.

Answer 2

ɢɪᴠᴇɴ:-

• √7-√5/√7+√5

ᴛᴏ Do:-

• Rationalisation of the denominator.

ꜱᴏʟᴜᴛɪᴏɴ:-

$\begin{lgathered}\begin{lgathered}\begin{lgathered}\begin{lgathered}:\implies\frak {\dfrac{ \sqrt{7} - \sqrt{5} }{ \sqrt{7} + \sqrt{5} } \times \dfrac{ \sqrt{7} - \sqrt{5} }{ \sqrt{7} - \sqrt{5}}} \\ \\\end{lgathered}\end{lgathered}\end{lgathered}\end{lgathered}$

$\begin{lgathered}\begin{lgathered}\begin{lgathered}\begin{lgathered}:\implies\frak {\dfrac{ \Big({ \sqrt{7} - \sqrt{5} \Big) }^{2} }{ {\Big( \sqrt{7}\Big)}^{2} + \Big({\sqrt{5} \Big)}^{2}}} \\ \\\end{lgathered}\end{lgathered}\end{lgathered}\end{lgathered}$

$\begin{lgathered}\begin{lgathered}\begin{lgathered}\begin{lgathered}:\implies\frak { \dfrac{7 + 5 - 2 \sqrt{35} }{7 - 5}}} \\ \\\end{lgathered}\end{lgathered}\end{lgathered}\end{lgathered}$

$\begin{lgathered}\begin{lgathered}\begin{lgathered}\begin{lgathered}:\implies\frak {\dfrac{7 + 5 - 2 \sqrt{35} }{7 - 5 }} \\ \\\end{lgathered}\end{lgathered}\end{lgathered}\end{lgathered}$

$\begin{lgathered}\begin{lgathered}\begin{lgathered}\begin{lgathered}:\implies\frak {\dfrac{12 - 2 \sqrt{35} }{2}} \\ \\\end{lgathered}\end{lgathered}\end{lgathered}\end{lgathered}$

$\begin{lgathered}\begin{lgathered}\begin{lgathered}\begin{lgathered}:\implies\frak {\dfrac{2(6 - \sqrt{35} )}{2} } \\ \\\end{lgathered}\end{lgathered}\end{lgathered}\end{lgathered}$

$\begin{lgathered}\begin{lgathered}\begin{lgathered}\begin{lgathered}:\implies\frak { \dfrac{ \cancel2(6 - \sqrt{35} )}{ \cancel2} } \\ \\\end{lgathered}\end{lgathered}\end{lgathered}\end{lgathered}$

$\begin{lgathered}\begin{lgathered}\begin{lgathered}\begin{lgathered}:\implies\frak { 6 - \sqrt{35} } \\ \\\end{lgathered}\end{lgathered}\end{lgathered}\end{lgathered}$

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ꜰᴏʀᴍᴜʟᴀᴇ ᴜꜱᴇᴅ:-

• ( a - b )² = a² + b² - 2ab

• ( a + b )( a - b ) = a² - b²